A292306 -id:A292306

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A265949

Expansion of Product_{k>=1} (1 + k^k*x^k).

+10
9

1, 1, 4, 31, 283, 3489, 50913, 890635, 17891170, 409850236, 10494427982, 297780829216, 9261266862273, 313453533534739, 11464487066049791, 450644378868285130, 18942868694407904729, 847930346323808122469, 40266107916200371331007, 2021842180288047801103956

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

FORMULA

a(n) ~ n^n * (1 + exp(-1)/n + ((1/2)*exp(-1) + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d + 1)*d^(k+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018

MAPLE

seq(coeff(series(mul((1+k^k*x^k), k=1..n), x, n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

MATHEMATICA

nmax=20; CoefficientList[Series[Product[(1+k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

PROG

(PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, (1+k^k*x^k))) \\ G. C. Greubel, Oct 31 2018

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

CROSSREFS

Cf. A023882, A292190, A292305, A292306.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Dec 19 2015

STATUS

approved

A292190

Sum of n-th powers of products of terms in all partitions of n into distinct parts.

+10
8

1, 1, 4, 35, 337, 11925, 371081, 49032439, 3545396034, 3416952655320, 749189363202730, 598250899004413536, 2383502427069445040595, 1729793152213690218766715, 131751643363739706679145099315, 271212858254426215135033141804302

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..98

FORMULA

a(n) = [x^n] Product_{k=1..n} (1 + k^n*x^k).

EXAMPLE

5 = 4 + 1 = 3 + 2. So a(5) = 5^5 + (4*1)^5 + (3*2)^5 = 11925.

MAPLE

b:= proc(n, i, k) option remember; (m->

`if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+

`if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)

end:

a:= n-> b(n$3):

seq(a(n), n=0..20); # Alois P. Heinz, Sep 11 2017

MATHEMATICA

nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 12 2017 *)

PROG

(PARI) {a(n) = polcoeff(prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}

CROSSREFS

Main diagonal of A292189.

Cf. A265949, A292167, A292194, A292305, A292306.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 11 2017

STATUS

approved

A292305

a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

+10
6

1, 1, 4, 54, 512, 9375, 186624, 4117715, 100663296, 3099363912, 100000000000, 3423740047332, 133741506723840, 5451751918660554, 244464150162276352, 11823135040283203125, 590295810358705651712, 31435129951680797038726, 1809934771463640728469504

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

FORMULA

a(n) = n^n * A000009(n).

a(n) ~ exp(Pi*sqrt(n/3)) * n^(n-3/4) / (4*3^(1/4)).

MATHEMATICA

nmax = 20; Table[SeriesCoefficient[Product[(1+n^k*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

Flatten[{1, Table[n^n*PartitionsQ[n], {n, 1, 20}]}]

CROSSREFS

Cf. A265949, A292190, A291698, A292306.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Sep 14 2017

STATUS

approved

A318247

a(n) = [x^n] Product_{k>=1} (1 + n!*x^k).

+10
1

1, 1, 2, 42, 600, 28920, 374285520, 128100273840, 131101518683520, 143354704247556480, 173401404266683545849388800, 2538767479410416957720411116800, 105287752487031026606448840363801600, 4510685217145833106538730603088118860800, 288804138719404983322786510403231912442931200

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..65

MATHEMATICA

nmax = 15; Table[SeriesCoefficient[Product[(1+n!*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

CROSSREFS

Cf. A077365, A265950, A292306, A292414, A318246.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Aug 22 2018

STATUS

approved

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